Given that \( S_1 \) and \( S_2 \) are identical springs, they each have the same spring constant \( k \). When both springs are attached, they are in parallel, so the effective spring constant \( k_{\text{eq}} \) is:

\[
k_{\text{eq}} = k + k = 2k
\]

The frequency \( f \) of oscillation for mass \( m \) with effective spring constant \( k_{\text{eq}} = 2k \) is:

\[
f = \frac{1}{2\pi} \sqrt{\frac{2k}{m}}
\]

If One Spring is Removed

If one spring is removed, only one spring with constant \( k \) is left. The new frequency \( f' \) becomes:

\[
f' = \frac{1}{2\pi} \sqrt{\frac{k}{m}}
\]

Ratio of New Frequency to Original Frequency

\[
\frac{f'}{f} = \frac{\frac{1}{2\pi} \sqrt{\frac{k}{m}}}{\frac{1}{2\pi} \sqrt{\frac{2k}{m}}} = \frac{\sqrt{\frac{k}{m}}}{\sqrt{\frac{2k}{m}}} = \frac{1}{\sqrt{2}}
\]

Thus:

\[
f' = \frac{f}{\sqrt{2}}
\]

Answer: \( f' = \frac{f}{\sqrt{2}} \)

Cutting a spring of constant \(k\) into 3 equal parts increases each part's spring constant to \(3k\). In a parallel connection, the equivalent constant is \(k_{\text{eq}} = 3k + 3k + 3k = 9k\). The new time period is \(T' = 2\pi\sqrt{\frac{m}{9k}} = \frac{T}{3}\).

The spring constant is \(k = \frac{F}{x} = \frac{10}{0.05} = 200\text{ N/m}\). The time period of the spring-mass system is \(T = 2pi \sqrt{\frac{m}{k}} = 2pi \sqrt{\frac{2}{200}} = 0.2pi \approx 0.628\text{ s}\).

The time period of a vertical spring-block system is \(T = 2\pi \sqrt{\frac{m}{k}}\), which is independent of gravity (g). So (A) is false. The equilibrium extension is \(x_{eq} = \frac{mg}{k}\). Since \(g_{moon} < g_{earth}\), then \(x_{eq,moon} < x_{eq,earth}\). So (R) is also false.

The time period of a spring-mass system is \(T = 2\pi \sqrt{m/k}\). This equation shows that the time period is independent of the acceleration due to gravity \(g\). Therefore, a spring block watch would function correctly in an orbiting satellite where effective \(g\) is zero. Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.

The periodic time of a spring is given by `\(T = 2\pi\sqrt{\frac{m}{k}}\)`.
A hard spring has a large spring constant `\(k\)`, which means a smaller `\(T\)`. A soft spring has a small `\(k\)`, hence a larger `\(T\)`. Both A and R are true, and R explains A.